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Operation meaning in set theory

When reading a set theorem, there is a part that: Defined a sequence $S_{n}$. Let $N = \inf\{n:S_{n}=0\}$ and let $X_{n} = S_{N \land n}$ I could not understand the operation $ \land$ in $S_{N \land ...
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0answers
15 views

Probability: Monkey drawing on paper

How can I calculate the probability of the following problem ? ; **suppose that a monkey without any previous experiences has a paper and a pen and draw with it on the paper ; What is the probability ...
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0answers
5 views

Derivation of Broyden's Method

I'm currently struggling with the drivation of the Broyden's method [1]. I get the point where the Jacobian $J$ is approximated via a (kind of) Secand method $A$ and has to fulfill the following ...
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0answers
4 views

Finite difference modeling for vibrations of a string. Simple case. Why is displacement increasing to infinity?

I'm trying to implement the finite difference modeling described here for a simple vibrating string: http://hplgit.github.io/num-methods-for-PDEs/doc/pub/wave/html/._wave001.html However, I can't make ...
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0answers
6 views

A proof that Mobius transformation preserve symmetry.

By definition, the points $z$ and $z^{\ast}$ are said to be symmetric with respect to the circle $C$ through $z_1,z_2,z_3$ if and only if $(z^{\ast},z_1,z_2,z_3)=\overline{(z,z_1,z_2,z_3)}$, where $(w,...
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0answers
9 views

Combinatorics, sequences

What is the number of non-increasing sequences of length n with elements from {0,1,2,⋯,n} with fixed first element k, k=0, 1, ... , n ? Consider the set of numbers from 1 to n. We construct non-...
2
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0answers
16 views

Show that $[x]+\left[x+\frac{1}{n}\right]+\left[x+\frac{2}{n}\right]+\cdots+\left[x+\frac{n-1}{n}\right]=[n x]$ [duplicate]

for any real number x and for any positive integer n show that $[x]+\left[x+\frac{1}{n}\right]+\left[x+\frac{2}{n}\right]+\cdots+\left[x+\frac{n-1}{n}\right]=[n x]$ My approach:- Let $\mathrm{x}=[\...
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0answers
5 views

Diamond Distribution in system K (Garson Modal Logic exercise 1.8)

I want to prove $\Diamond (P \lor Q) \Rightarrow \Diamond P \lor \Diamond Q$ It was a biconditional, but I have proved the other one. Thanks for the answer. Please use Garson's method. Thanks. I am ...
1
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2answers
19 views

Proving that $\vec{r'}(t)$ is orthogonal to $\vec{r''}(t)$

With a given nonzero vector $\vec{r}(t)$, how do I that $\vec{r'}(t)$ is orthogonal to $\vec{r''}(t)$? The length ($||\vec{r'}(t)||$ is constant.) This is what I have tried so far. Let $\vec{r}(t)= &...
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0answers
4 views

Bounding distance between intersection point and minimizer of 2 monotonic functions

Given a non decreasing function $f:[0,1]\rightarrow \mathbb{R}_{\geq 0}$, and a non increasing function $g:[0,1]\rightarrow \mathbb{R}_{\geq 0}$ that intersect at $x^{\bot}$. And given $x^* = \text{...
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0answers
9 views

Finding Coefficient From Curl

$$E= (ax+2y)i_x + (2-2y)i_y + (y-z)i_z$$ is the curl where ($i_x,i_y,i_z$) are the components. So, what is $a$?
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1answer
14 views

Simplifying $\frac{1}{\sqrt{x-i}}\left\{1-2 \sum_{n=1}^{\infty}(-1)^{n-1} \exp \left(-\pi n^{2}\left(\frac{1}{x-i}+i\right)\right)\right\}$

This is related to another recent question of mine. Considering that $$ \psi(x)=\sum_{n=1}^{\infty} \exp \left(-n^{2} \pi x\right) $$ has the finctional equation $$ \frac{1+2 \psi(x)}{1+2 \psi(1 / x)}=...
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0answers
22 views

A question on Hilbert space

Let $Y$ be a subspace closed of Hilbert space $H$. Why for any $x\in H$, there exist unique decomposition $x=y+z$ such that $y\in Y$ and $z\in Y^{\perp }$.
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2answers
14 views

Enumerate the elements of a quotient Gaussian Integers ring

I want to find an extensive list of all the elements of the quotient ring $Z[i]/(3+i)$. Since the Gaussian integers are an euclidean domain with euclidean function $N(a+bi)=a^2+b^2$ the representative ...
0
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1answer
3 views

find the range of a added parameter in linear programming problem remains the optimal solution

How can I find the range of an added parameter in the linear programming problem which preserve the original optimal solution remaining to be the optimal solution? such as: $$ Max \space\space\space ...
0
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1answer
28 views

Write $f(x) = \displaystyle \int_{-2}^{x}t|t-1|dt$ without the sign of integral.

Here is my attempt: $\displaystyle \int_{-2}^{x}t|t-1|dt$ =$\begin{cases} \displaystyle\int_{-2}^{x}t^2-tdt, \; t \ge 1 \\\displaystyle \int_{-2}^{x}-t^2+tdt, \; t <1\end{cases}$ =$\begin{cases} \...
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3answers
22 views

Find a basis for S $\cap$ T. Also find its dimension. Conditions are as following.

Let $S=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\in\Bbb{R}:a_{1}+a_{2}+a_{3}+a_{4}=0\right\}$ And $T=\left\{\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\in\Bbb{R}:a_{1}-a_{2}+a_{3}-a_{4}=0\right\}$ ...
2
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0answers
16 views

Are there more learnable but undecidable cases except the halting problem

In the ICML 1992 paper, On the Learnability of the Uncomputables, by Richard Lathrop, he proved that halting problem is learnable in a probabilistic learnability. So except halting problem, are there ...
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1answer
29 views

Showing a function is constant - Complex analysis

I am trying to solve the following problem. $f(z)=u(x,y)+iv(x,y)$ is an analytic function in $D$ ($D$ is connected and open). If $u, v$ fulfill the relation $G(u(x,y), v(x,y))= 0 $ in $D$ for some ...
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2answers
11 views

Domain of the natural logarithm of a factorisable quadratic polynomial

Suppose I have a function that is the natural logarithm of a quadratic polynomial, which can be factorised: $$ f(x) = \ln(x^2+2x-8)$$ The domain of $f$ is $x \lt -4$ and $x \gt 2$. However, if I ...
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0answers
8 views

Gradient and Hessian of squared Frobenius norm

I want to find the Gradeint and Hessian of the following function, $F(\mathbf{S}) = \frac{1}{2}\Vert \mathbf{M} - \mathbf{K_2SK_1^T}\Vert _F^2+\frac{1}{2}\Vert\mathbf{S}\Vert_F^2$. My try: Using trace ...
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3answers
38 views

What are some other ways of solving this limit?

I have come across many limit questions of the form $0^0$ For instance , here is one example $$y=\lim_{x \to 0^+} (2\sin(\sqrt x) + \sqrt x\sin{1\over x})^x$$ In this example if I take logarithm on ...
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1answer
12 views

Help with finding lower bound of a binomial expansion

I have to find the lower bound of $1 - [1 - (2^{-m/2} - 2^{-m})]^p$. I should get the answer as $p*2^{-m/2} - p(p+1)*2^{-m}$. I tried to do binomial expansion but I am stuck. Any help is appreciated.
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0answers
11 views

Question about $T-$cyclic subspace generated

$\textbf{Definition:}$ Let $V$ be a finite dimensional vector space over a field $F$ and let $T:V \to V$ be a linear operator. If $v$ is a vector in $V$, the $T-$cyclic subspace generated by $v$ is ...
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0answers
14 views

Expected value of red/blue ball game

There are 4 red balls and 3 blue balls in an urn. Picking red balls gives you 1 dollar and picking blue balls means you have to pay 1 dollar. You can stop picking balls at any point. What is the ...
2
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1answer
17 views

$[G:K]=[G:H][H:K]$ for arbitrary group in ZF

If $G$ is any group, and $H$ and $K$ are subgroups such that $H\supseteq K$, then can we prove that $[G:K]=[G:H][H:K]$ only using ZF, where $[A:B]$ is the cardinality of the set $\{Bx:x\in A\}$, where ...
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2answers
25 views

Why can't the “chain rule” of derivatives be used to differentiate 3sin(x)?

My understanding is that functions of the form $f(g(x))$ can be differentiated using the "chain rule", where $$\frac{d}{dx}f(g(x)) = f'g(x) \cdot g'(x)$$ I was trying to apply that logic to ...
2
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0answers
16 views

On the Borel Class

Suppose that $T$ is a locally compact Hausdorff space. We can consider the Borel class in this space, namely, the $\sigma$-algebra generated by open sets of $T$. The author wrote that, if $E\subseteq ...
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0answers
11 views

Product of operator with its adjoint is self-adjoint

Suppose $A$ is a densely defined closed operator. Show that $A^*A$ (with domain $D(A^*A)=\{\psi\in D(A)|A\psi\in D(A^*)\}$) is self-adjoint. Let $\psi\in D(A^*A)$ and observe that $A^*A$ is ...
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0answers
9 views

Bounds on measures converted to bounds on densities

Suppose that $P$ and $Q$ are two probability measures such that $$ P(A) \leq cQ(A) + c' $$ where $$ c' = \sup_{A}\Big( P(A) - cQ(A) \Big). $$ When $c' = 0$, it is easy to see that the Radon-Nikodym ...
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0answers
9 views

How to get values of estimators in the SRF?

I was reading the Econometrics textbook by Damodar Gujarati, and in the book, he develops the sample regression function using the following equations. $\sum Y_{i} = n \beta_{1} + \beta_{2}\sum X_{i}$ ...
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0answers
15 views

How would I write for all x and for all y in the same domain D with first-order logic syntax? [closed]

Is it something like this? $$ (\forall x \forall y) \in D ... $$
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1answer
16 views

What's the proof for this summation with dependent variables?

Given $0\leqslant i<j\leqslant n$ $\sum_{i=0}^n\sum_{j=0}^n i = \sum_{i=0}^n(n+1)*i$ It seems to work like a nested loop and gave the right answer when substituting n for any number but I don't ...
1
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1answer
30 views

Prove that $\prod_{i=1}^{n}i^{n+1-i}=\prod_{i=1}^{n}(n+1-i)^i$ by induction.

I'm trying to prove that $$\prod_{i=1}^{n}i^{n+1-i}=\prod_{i=1}^{n}(n+1-i)^i$$ by induction. I've proved it already a different way, but I'm interested to see what a proof by induction would look like ...
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0answers
16 views

How to find coefficient from gradient vector?

If $E= (3x+2y)I_x + (2-ax+2z)I_y + (2y-z)I_z$ is a gradient of vector. Find $a$ where $$Ix,Iy,Iz$$ are vector components like $i,j,k$
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0answers
4 views

Reconstruction of a binary vector from any two rows of a binary matrix

Let $ a = (a_1,a_2, \ldots,a_{10})\in \{ 0,1\}^{10 \times 1}$ be a binary column vector of length $10$. How to find $x_{i,j} \in \{ 0,1\}^{1 \times 10}$, $i\in \{1,2,3,4,5\}$, $j \in \{1,2,3\}$ (...
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1answer
31 views

Can someone explain to me mathematical models in matrix notation?

I am trying to understand multivariate statistical models, but unfortunately my linear algebra is poor. I saw this example of converting three models into matrix notation: Formulating regression ...
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0answers
12 views

Would any two consecutive extreme points on a convex hull can be linearly projected still being extreme?

Given two extreme points on a convex hull, if the straight line connects them is on the boundary , they are consecutive. Would there always exits a linear projection into lower dimension such that ...
0
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0answers
11 views

Doubts about a density function X-Y, just need a little more information [duplicate]

I have this problem Let 𝑋 and 𝑌 be random variables that have a joint density function given by, $$f_{X,Y}(x,y)=\left\{ \begin{array}{1} 8xy, \space 0 \leqslant x<y \leqslant1 \\ 0, \space \text{...
0
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0answers
17 views

Find the integral of $\int\limits_0^{2\pi } {Q\left( {f(\theta )} \right)Q\left( {g(\theta )} \right)d\theta } $?

I am trying to find the integral of the following function: $\int\limits_0^{2\pi } {Q\left( { - (e*\cos (a + \theta )*sqrt(x) + g} \right)*Q\left( { - (f*\sin (a + \theta )*sqrt(x) + h} \right)d\theta ...
0
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1answer
16 views

Can I determine if the result of a pairing function is the from the inverse of a given pair?

For example, say you had the following results: f(a, b) = c f(b, a) = d Is there a pairing function that would allow for determining that c is sort of the "inverse" of d without de-pairing ...
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0answers
12 views

On bounds for the deficiency of $m^2$, where $p^k m^2$ is an odd perfect number with special prime $p$ - Part II

(Note: This question is a sequel to this earlier post.) Hereinafter, call a number $N$ perfect if $N$ satisfies $\sigma(N)=2N$, where $$\sigma(x)=\sum_{d \mid x}{d}$$ is the sum of divisors of the ...
0
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1answer
13 views

How can we find the crossing point of two lines with a prescribed angle?

In the 3D space, we have two given points of $P$ and $Q$. Line $A$ passes through the point $P$ and whose angle with the x-axis is $\theta$ and with the z-axis $\phi$. Line $B$ passes through the ...
0
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0answers
9 views

For $P_{i}=X(X-1) \ldots(X-i+1)$, decompose $P_{i}^{*}$ in terms of $f_{i}: P \mapsto P(i)$.

I know $P_{i}=X(X-1) \ldots(X-i+1)$ is a basis of $\mathbb{K}_{n}[X]$ for i=0 to n and $f_{i}: P \mapsto P(i)$ is a basis of $(\mathbb{K}_{n})^{*}$ but I am having serious difficulty representing the $...
0
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0answers
32 views

Rudin Proposition 1.18(c)

I'm trying to understand Rudin's proof of part (c) of Propositon 1.18. These statements concern ordered fields. 1.18 (c) If $x < 0$ and $y < z$ then $xy > xz$. The results Rudin suggests ...
6
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0answers
38 views

Infinitely many common prime divisors

By Zsigmondy's theorem, there are infinitely many prime divisors of $2^{2^n}-1$. That is, the set $$A=\{p \text{ is a prime}: p\mid 2^{2^n}-1 \text{ for some }n\in\Bbb{N}\}$$ is infinite. Also, as ...
1
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1answer
19 views

Conditions for symmetric, Toeplitz $\mathbf{M}$ with nonnegative elements to have inverse with nonnegative elements

Problem Suppose we have symmetric, Toeplitz matrix $\mathbf{M}$ such that $$ \mathbf{M} = \begin{bmatrix} m_0 & m_1 & m_2 & m_3 & \cdots &m_{n-1} \\ m_1 & m_0 & m_1 & ...
-1
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0answers
21 views

Find the minimum possible perimeter of triangle formed

I have a triangle which is having sides AB=10, BC=14, AC=16 and M is the midpoint of BC. Various lines can be drawn through M, cutting AB(possibly extended) at P and AC(possibily extended) at Q. ...
1
vote
1answer
15 views

Converting between bound on probability measures and densities

Suppose that $P$ and $Q$ are two probability measures on the same probability space with $P(A) \leq c Q(A)$ for each (measurable) set $A$. Is it true that $dP/dQ$ is then bounded by $c$ $P$-almost ...
1
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0answers
13 views

gradient computation for neural ODEs

I was reading the paper on Neural ODEs (here) and was wondering if anyone could offer some insight on calculation of the gradient of the loss function. If we are only considering 2 time points, $t_0,...

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